EigenChris: Tensor Calculus & Differential Geometry
Visual introduction to tensors, covariant derivatives, and Riemannian geometry
About This Series
EigenChris's tensor calculus series is widely regarded as the best visual introduction to tensor mathematics and differential geometry available online. These 43 lectures (16 on tensors + 27 on tensor calculus) build from basic definitions through the Riemann curvature tensor and Einstein field equations.
🎨 Visual Excellence
Beautiful diagrams and geometric intuition throughout
📊 Progressive Build
From vectors through differential forms to curvature
🔗 GR Foundation
Essential preparation for general relativity
Why Study Tensor Calculus?
For General Relativity
- • Metric Tensor: Spacetime geometry g_μν
- • Christoffel Symbols: Connection and geodesics
- • Riemann Tensor: Spacetime curvature R^ρ_σμν
- • Einstein Equations: G_μν = 8πG T_μν
- • Foundation for black holes, cosmology, gravitational waves
Beyond Physics
- • Differential Geometry: Modern geometry of manifolds
- • Gauge Theory: Fiber bundles and connections
- • String Theory: Higher-dimensional geometry
- • Machine Learning: Information geometry, natural gradients
- • Pure mathematics and modern geometry
Course Structure
Part I: Tensors for Beginners (0-16)
- • Vector and covector spaces
- • Transformation rules (contravariant/covariant)
- • Metric tensor and raising/lowering indices
- • Bilinear forms and tensor products
- • General tensor definition
Part II: Tensor Calculus (0-26)
- • Jacobian and coordinate transformations
- • Differential forms and integration
- • Covariant derivative and parallel transport
- • Geodesics and Christoffel symbols
- • Riemann, Ricci, and scalar curvature
Part I: Tensors for Beginners (17 lectures)
Motivation
Why study tensors? Applications in physics, engineering, and mathematics
Video Lecture
Tensors For Beginners (-1): Motivation
Introduction and motivation for tensor theory
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Tensor Definition
What is a tensor? Multilinear maps and transformation laws
Video Lecture
Tensors for Beginners 0: Tensor Definition
Formal definition of tensors as multilinear maps
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Forward and Backward Transformations (REMAKE)
Coordinate transformations and their inverses, Jacobian matrices
Video Lecture
Tensors for Beginners 1: Forward and Backward Transformations
Understanding coordinate transformations
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Vector Transformation Rules
Contravariant vectors: V'^i = (∂x'^i/∂x^j) V^j
Video Lecture
Tensors for Beginners 3: Vector Transformation Rules
How vectors transform under coordinate changes
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Lectures 4-6: Covectors (Dual Vectors)
Video Lecture
Tensors for Beginners 4: What are Covectors?
Introduction to one-forms and dual spaces
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensors for Beginners 5: Covector Components
How to write covectors in component form
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensors for Beginners 6: Covector Transformation Rules
Covariant transformation: ω'_i = (∂x^j/∂x'^i) ω_j
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Lectures 7-8: Linear Maps as Tensors
Video Lecture
Tensors for Beginners 7: Linear Maps
Linear transformations as (1,1) tensors
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensors for Beginners 8: Linear Map Transformation Rules
How (1,1) tensors transform
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Lectures 9-10: The Metric Tensor
Video Lecture
Tensors for Beginners 9: The Metric Tensor
g_ij: measuring distances and angles
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensors for Beginners 10: Bilinear Forms
General (0,2) tensors and symmetric bilinear forms
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Lectures 11-14: Tensor Products & General Tensors
Video Lecture
Tensors for Beginners 11: Linear maps are Vector-Covector Pairs
Decomposing (1,1) tensors
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensors for Beginners 12: Bilinear Forms are Covector-Covector pairs
(0,2) tensor decomposition
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensors for Beginners 13: Tensor Product vs Kronecker Product
Distinguishing ⊗ from matrix Kronecker product
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensors for Beginners 14: Tensors are general vector/covector combinations
General (r,s) tensors T^(i₁...iᵣ)_(j₁...jₛ)
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Lectures 15-16: Tensor Spaces & Index Gymnastics
Video Lecture
Tensors for Beginners 15: Tensor Product Spaces
Structure of tensor spaces T^r_s(V)
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensors for Beginners 16: Raising/Lowering Indexes
Musical isomorphisms: ♯ and ♭ operators
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Part II: Tensor Calculus (27 lectures)
This series builds on tensors to develop calculus on manifolds: differentiation, integration, covariant derivatives, parallel transport, geodesics, and curvature. Essential for general relativity!
Lectures 0-2: Introduction & Multivariable Calculus
Video Lecture
Tensor Calculus 0: Introduction
Course overview and prerequisites
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Video Lecture
Tensor Calculus 1: Multi-variable Calculus Review
Partial derivatives and gradients
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 2: Cartesian/Polar Coordinates, and Basis Vectors
Coordinate systems and basis vectors
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Lectures 3-5: The Jacobian & Contravariant Derivatives
Video Lecture
Tensor Calculus 3: The Jacobian
Jacobian matrix ∂x^i/∂u^j
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 4: Derivatives are Vectors
∂/∂x^i forms a vector basis
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 5: Derivative Transformation Rules
Contravariance of ∂/∂x^i
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 5.1: Derivative Operators are Vectors Discussion
Optional deep dive
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Lectures 6-10: Differential Forms & Integration
Video Lecture
Tensor Calculus 6: Differential Forms are Covectors
dx^i as covector basis
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 7: Covector Field Components
Writing ω = ω_i dx^i
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 8: Covector Field Transformation Rules
Covariance of differential forms
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 9: Integration with Differential Forms
∫ω over manifolds
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 10: Integration Examples
Concrete integration problems
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Lectures 11-14: Metric Tensor & Gradient
Video Lecture
Tensor Calculus 11: Metric Tensor and Arc Lengths (flat)
ds² = g_ij dx^i dx^j in flat space
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 12: Metric Tensor in Curved Spaces
Curved manifold metrics
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 13: Gradient vs 'd' operator
Exterior derivative vs gradient
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 14: Gradient explanation + examples
grad f = g^(ij) ∂f/∂x^j
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Lectures 15-16: Geodesics & Christoffel Symbols (Extrinsic)
Video Lecture
Tensor Calculus 15: Geodesics and Christoffel Symbols
Shortest paths and connection coefficients
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Video Lecture
Tensor Calculus 16: Geodesic Examples
Great circles on spheres
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Lectures 17-20: Covariant Derivative & Parallel Transport
Video Lecture
Tensor Calculus 17: Covariant Derivative (flat space)
∇_j V^i in flat coordinates
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 17.5: Covariant Derivative Components
Optional component-based approach
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 18: Covariant Derivative (extrinsic)
Connection via embedding
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 19: Covariant Derivative (Intrinsic)
Intrinsic definition, geodesic equation
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 20: Abstract Covariant Derivative
Levi-Civita connection axioms
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Lecture 21: Lie Bracket, Flow, Torsion
Video Lecture
Tensor Calculus 21: Lie Bracket, Flow, Torsion Tensor
[X,Y] and torsion-free connections
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Lectures 22-26: Riemann Curvature Tensor ⭐
The Climax: The Riemann tensor R^ρ_σμν captures all curvature information. Essential for Einstein's field equations!
Video Lecture
Tensor Calculus 22: Riemann Curvature Tensor Geometric Meaning
Holonomy and geodesic deviation
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Video Lecture
Tensor Calculus 23: Riemann Curvature Components & Symmetries
20 independent components in 4D
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Video Lecture
Tensor Calculus 24: Ricci Tensor Geometric Meaning
Sectional curvature R_μν
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 25: Ricci Tensor/Scalar (Volume Form)
Volume distortion and scalar R
💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.
Video Lecture
Tensor Calculus 26: Ricci Tensor/Scalar Properties
Contractions and Einstein tensor
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How to Use This Series
📚 Before General Relativity
Watch all 43 lectures before starting GR. This builds the essential mathematical foundation.
- • Weeks 1-2: Tensors for Beginners (0-16)
- • Weeks 3-4: Tensor Calculus basics (0-14)
- • Week 5: Covariant derivative (15-21)
- • Week 6: Curvature (22-26)
- • Then: Ready for Einstein equations!
🔄 Alongside Mathematics
Use these videos to supplement the Mathematics course
- • Visual intuition for abstract concepts
- • Concrete examples and computations
- • Geometric pictures of tensors
- • Bridge to physics applications
Integrated Study Path
Linear Algebra: Mathematics: Tensor Calculus
EigenChris Tensors: Lectures 0-16 (visual geometric intuition)
EigenChris Calculus: Lectures 0-26 (differential geometry)
General Relativity: Now ready for Einstein's field equations!
Additional Resources
- • Tensors Playlist:Tensors for Beginners (17 videos)
- • Tensor Calculus Playlist:Tensor Calculus (27 videos)
- • Creator:@eigenchris on YouTube
- • Other Series:Spinors for Beginners, General Relativity